Sampling in Action: The M&M Challenge

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<a href="https://psychmethods.github.io/coursenotes/" target="_blank">Methods in Psychological Research</a>
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# Sampling in Action: The M&M Challenge

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## Roadmap

- M&M Sampling Activity
- Analysis and Discussion
- Advanced Sampling Concepts

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## Sampling in Action

Sampling involves selecting a subset of a population to estimate characteristics of the whole.

This session covers:

- Sample Size Effects: How different sizes influence accuracy.
- Variability in Sampling: The impact on data reliability.
- Estimation Techniques: Methods for deriving population parameters from samples.

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## M&M Sampling Activity

- Objective: Demonstrate sampling principles using M&M's
  - Hands-on experience with data collection and analysis
  
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- Materials:
  - Small packages of plain M&M's (one per student)
  - Napkins for sorting

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## M&M Sampling Procedure

*Steps in the activity*

- Distribute M&M packages and materials
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- Sort M&M's by color on napkins
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- Record frequency of each color
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- Calculate percentages for each color
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- Hypothesize population color distribution
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- Form pairs to pool data
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- Pool data for entire class
    - using google sheets (and some R magic)

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# Data Collection and Analysis

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```r
set.seed(123) # For reproducibility

# Define the number of students and colors
students <- c("Tukey", "Gauss", "Noether", "Fisher", 
              "Bayes", "Pearson", "Student", 
              "Fiducial", "Neyman", "Cochran")
colors <- c("Blue", "Brown", 
            "Green", "Red", "Yellow")

# Simulate the total number of M&Ms for each student
total_mms <- sample(15:20, 
                    length(students), replace = TRUE)

# Simulate the counts of each color for each student
color_counts <- replicate(length(colors), 
                          sample(1:total_mms, 
                                 length(students), replace = TRUE))

# Create the dataframe
df_syn <- data.frame(Name = students, 
                     color_counts)
colnames(df_syn)[-1] <- colors

# Calculate the percentages
df_syn <- df_syn %>%
  mutate(Total = rowSums(across(Blue:Yellow))) %>%
  mutate(Blue_perc = Blue / Total * 100,
         Brown_perc = Brown / Total * 100,
         Green_perc = Green / Total * 100,
         Red_perc = Red / Total * 100,
         Yellow_perc = Yellow / Total * 100)

# Reshape the data to long format
df_long_syn <- df_syn %>%
  pivot_longer(cols = c(Blue_perc, Brown_perc, Green_perc, Red_perc, Yellow_perc),
               names_to = "Color",
               values_to = "Percentage")

# Plotting the data
stacked_plot <- df_long_syn %>%
  ggplot(aes(x = Name, y = Percentage, fill = Color)) +
  geom_col(position = "stack") +
  labs(title = "M&M Color Distribution by Student",
       x = "Student",
       y = "Percentage") +
  scale_fill_manual(values = c("Blue_perc" = "blue",
                               "Brown_perc" = "brown",
                               "Green_perc" = "green",
                               "Red_perc" = "red",
                               "Yellow_perc" = "yellow"),
                    labels = c("Blue_perc" = "Blue",
                               "Brown_perc" = "Brown",
                               "Green_perc" = "Green",
                               "Red_perc" = "Red",
                               "Yellow_perc" = "Yellow")) +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 45, hjust = 1))

# Overall distribution of M&Ms
overall_distribution <- df_syn %>%
  select(Blue, Brown, Green, Red, Yellow) %>%
  summarise(across(everything(), sum)) %>%
  pivot_longer(cols = everything(), names_to = "Color", values_to = "Count")

overall_plot <- overall_distribution %>%
  ggplot(aes(x = Color, y = Count, fill = Color)) +
  geom_col() +
  labs(title = "Overall M&M Color Distribution",
       x = "Color",
       y = "Total Count") +
  scale_fill_manual(values = c("Blue" = "blue",
                               "Brown" = "brown",
                               "Green" = "green",
                               "Red" = "red",
                               "Yellow" = "yellow")) +
  theme_minimal()

# Display both plots
library(gridExtra)
grid.arrange(stacked_plot, overall_plot, ncol = 2)
```

]

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# Inputting Student Data

```
## Warning: Removed 84 rows containing missing values or values outside the
## scale range (`geom_col()`).
```

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# Sample Size Effects

- Compare estimates:
    - Individual samples
    - Paired samples
    - Class-wide sample
- Discuss accuracy improvements with larger samples

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# Example (Hypothetical data)

```r
# Individual sample
individual <- c(20, 15, 25, 10, 20, 10)
names(individual) <- c("Blue", "Brown", "Green", "Orange", "Red", "Yellow")

# Class-wide sample
class_wide <- c(24, 13, 16, 20, 13, 14)
names(class_wide) <- c("Blue", "Brown", "Green", "Orange", "Red", "Yellow")

# Compare
data.frame(Individual = individual, Class_Wide = class_wide)
```

```
##        Individual Class_Wide
## Blue           20         24
## Brown          15         13
## Green          25         16
## Orange         10         20
## Red            20         13
## Yellow         10         14
```

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# Visualization

```r
barplot(rbind(individual, class_wide), beside = TRUE, 
        col = c("lightblue", "darkgrey"),
        legend.text = c("Individual", "Class-wide"),
        main = "M&M Color Distribution: Individual vs Class-wide")
```

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# Advanced Sampling Concepts

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# Relating to Sampling Methods

- Simple random sampling
    - Each M&M package as a random sample
- Stratified sampling
    - If we sorted M&M bags by production date

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# Potential Biases

- Production process biases
    - Color distribution variations between factories
- Relating to survey sampling biases
    - Non-response bias
    - Selection bias

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# Importance of Representative Samples

- What if we only sampled from one factory?
- Implications for psychological research
    - Generalizing from sample to population

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# Wrapping Up...

<br><br>
![](../img/mms.jpg)

```